Theorem bd0 17055

Description: A formula equivalent to a bounded one is bounded. See also bd0r 17056. (Contributed by BJ, 3-Oct-2019.)

Hypotheses

Ref	Expression
bd0.min	⊢ BOUNDED 𝜑
bd0.maj	⊢ (𝜑 ↔ 𝜓)

Assertion

Ref	Expression
bd0	⊢ BOUNDED 𝜓

Proof of Theorem bd0

Step	Hyp	Ref	Expression
1		bd0.min	. 2 ⊢ BOUNDED 𝜑
2		bd0.maj	. . 3 ⊢ (𝜑 ↔ 𝜓)
3	2	ax-bd0 17044	. 2 ⊢ (BOUNDED 𝜑 → BOUNDED 𝜓)
4	1, 3	ax-mp 5	1 ⊢ BOUNDED 𝜓

This proof depends on syntax axioms:   ↔ wb 105  BOUNDED wbd 17043

This proof depends on axioms:  ax-mp 5  ax-bd0 17044

This theorem is used by:  bd0r  17056  bdth  17062  bdnth  17065  bdnthALT  17066  bdph  17081  bdsbc  17089  bdsnss  17104  bdcint  17108  bdeqsuc  17112  bdcriota  17114  bj-axun2  17146